Fixed-Parameter Algorithms for the Kneser and Schrijver Problems

TL;DR

The paper advances the study of fixed-parameter tractability for the computational Kneser and Schrijver problems by presenting randomized algorithms that locate monochromatic edges in $K(n,k)$ and $S(n,k)$ with an oracle coloring using $n-2k+1$ colors, achieving running time $n^{O(1)}\cdot k^{O(k)}$. Central to the approach is an element-elimination technique augmented by stability results for intersecting families (EKR, Hilton–Milner, and Dinur–Friedgut), which enables shrinking the ground set to a manageable size while preserving a guaranteed monochromatic edge with high probability. The work also establishes a polynomial-time randomized Turing kernelization to $O(k^4)$ ground-set instances for the Schrijver problem (and $O(k^3)$ for Kneser in a refined analysis), and provides a simple deterministic algorithm for the Kneser problem via Schrijver subgraphs. Furthermore, the paper connects these problems to the Agreeable-Set problem, deriving efficient algorithms for instances where the number of agents is not too small relative to the number of items, and constructing reductions that place Agreeable-Set at least as hard as Kneser with subset queries. Overall, the results illuminate parameterized pathways to total-search problems tied to topological colorings, with implications for kernelization, complexity within PPA, and related resource-allocation problems.

Abstract

The Kneser graph $K(n,k)$ is defined for integers $n$ and $k$ with $n \geq 2k$ as the graph whose vertices are all the $k$-subsets of $[n]=\{1,2,\ldots,n\}$ where two such sets are adjacent if they are disjoint. The Schrijver graph $S(n,k)$ is defined as the subgraph of $K(n,k)$ induced by the collection of all $k$-subsets of $[n]$ that do not include two consecutive elements modulo $n$. It is known that the chromatic number of both $K(n,k)$ and $S(n,k)$ is $n-2k+2$. In the computational Kneser and Schrijver problems, we are given an access to a coloring with $n-2k+1$ colors of the vertices of $K(n,k)$ and $S(n,k)$ respectively, and the goal is to find a monochromatic edge. We prove that the problems admit randomized algorithms with running time $n^{O(1)} \cdot k^{O(k)}$, hence they are fixed-parameter tractable with respect to the parameter $k$. The analysis involves structural results on intersecting families and on induced subgraphs of Kneser and Schrijver graphs. We also study the Agreeable-Set problem of assigning a small subset of a set of $m$ items to a group of $\ell$ agents, so that all agents value the subset at least as much as its complement. As an application of our algorithm for the Kneser problem, we obtain a randomized polynomial-time algorithm for the Agreeable-Set problem for instances with $\ell \geq m - O(\frac{\log m}{\log \log m})$. We further show that the Agreeable-Set problem is at least as hard as a variant of the Kneser problem with an extended access to the input coloring.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Theorem 2.2: LovaszKneserSchrijverKneser78
• Definition 2.3
• Theorem 2.4: Hilton--Milner Theorem HM67
• Remark 2.5